Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Condensed matter

Universality classes for the dynamic surface critical behavior of systems with relaxational dynamics

Hans Werner Diehl

Published Jan 15, 1994• 30 citations
Publisher

Abstract

We investigate the problem of which universality classes of dynamic surface critical behavior exist for semi-infinite systems whose dynamic bulk critical behavior and static surface critical behavior are representative of a given dynamic bulk universality class and a given static surface universality class. To this end systems whose bulk dynamics are described by either model B or model A of Halperin, Hohenberg, and Ma are considered, where the dynamics are allowed to be modified near the surface as follows: In the case of model B, surface terms that locally break the conservation law for the order-parameter density \ensuremath{\varphi} are allowed; in the case of model A, \ensuremath{\varphi} is assumed to be locally conserved near the surface. Semi-infinite field-theory models are constructed that are (i) compatible with the requested bulk dynamics and the requirements of causality, detailed balance, and relaxation to thermal equilibrium and (ii) minimal in the sense that no redundant or irrelevant surface terms are included. The boundary conditions to which the surface terms of the action correspond are worked out. For model B it is shown that nonconservative surface terms are relevant. Their strength can be parametrized by a coupling constant c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$\ensuremath{\ge}0 whose inverse plays the role of an extrapolation length for the auxiliary (Martin-Siggia-Rose) field \ensuremath{\varphi}\ifmmode \tilde{}\else \~{}\fi{}. Thus two distinct semi-infinite extensions of model B exist---one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$>0 called model ${\mathit{B}}_{\mathit{A}}$, and one with c${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}$=0 called model ${\mathit{B}}_{\mathit{B}}$---and each static surface universality class splits up into two dynamic surface universality classes.The static and dynamic surface critical exponents of these dynamic surface universality classes as well as the crossover exponent associated with the nonconservative surface term are shown to be expressible in terms of static bulk and surface exponents. Model ${\mathit{B}}_{\mathit{A}}$ and its results should apply, e.g., to uniaxial ferromagnets whose rotational spin symmetry is broken in the vicinity of a surface plane. For model A the considered local modification of the dynamics near the surface is irrelevant in the renormalization-group sense and hence does not lead to new dynamic surface universality classes.

DOI
10.1103/physrevb.49.2846

Related papers

Critical Adsorption in Systems with Weak Surface Field: The Renormalization-Group Approach

A. Ciach, Uwe Ritschel

We study the surface critical behavior of semi-infinite systems belonging to the bulk universality class of the Ising model. Special attention is paid to the local behavior of experimentally relevant quantities such as the order parameter and the correlation function in the crossover regimes between different surface universality classes, where the surface field h1 and the temperature enhancement c can induce additional macroscopic length scales. Starting from the field-theoretical φ4 model and employing renormalization-group improved perturbation theory (ϵ-expansion), explicit results for the local behavior of the correlation and structure function (zero-loop) and the order parameter (one-loop) are derived. Supplementing earlier studies that focused on the special transition, we here pay particular attention to the situation where a large c suppresses the tendency to order in the surface (ordinary transition) but a surface field h1 generates a small surface magnetization m1. Our results are in good agreement with recent phenomenological considerations and Monte Carlo studies devoted to similar questions and, combined with the latter, provide a much more detailed understanding of the local properties of systems with weak surface fields.

•Aug 17, 2016•12 citations

Universal Short-Time Behavior in Critical Dynamics near Surfaces

Uwe Ritschel, Peter Czerner

We study the time evolution of classical spin systems with purely relaxational dynamics, quenched from $T\ensuremath{\gg}{T}_{c}$ to the critical point, in the semi-infinite geometry. Shortly after the quench, as in the bulk, a nonequilibrium regime governed by universal power laws is also found near the surface. We show for "ordinary" and "special" transitions that the corresponding critical exponents differ from their bulk values, but can be expressed via scaling relations in terms of known bulk and surface exponents. To corroborate our scaling analysis, we present perturbative ($\ensuremath{\epsilon}$-expansion) and Monte Carlo results.

arXiv:cond-mat/9510057•Physical Review Letters•Nov 20, 1995•41 citations

Surface critical behavior in the presence of linear or cubic weak surface fields

Hans Werner Diehl, A. Ciach

The effects of weak symmetry-breaking surface fields on the surface critical behavior of semi-infinite systems with a scalar order parameter are investigated near a bulk critical point. Previous field-theoretic analyses in 4-\ensuremath{\epsilon} dimensions are generalized by incorporating an additional surface term ${\mathit{w}}_{0}$${\mathrm{\ensuremath{\varphi}}}_{\mathit{s}}^{3}$ in the Hamiltonian. The mapping of microscopic models of the lattice-gas type on a continuum field theory is reconsidered, and we conclude that the cubic surface field ${\mathit{w}}_{0}$ generically does not vanish both in the case of an Ising ferromagnet in an applied magnetic field H (where ${\mathit{w}}_{0}$\ensuremath{\sim}H as H\ensuremath{\rightarrow}0) as well as in the case of critical adsorption of fluids onto a wall or an interface. Explicit results of a perturbative renormalization-group analysis are given, which show that a fixed point with nonvanishing cubic surface field ${\mathit{w}}^{\mathrm{*}}$=O(\ensuremath{\surd}\ensuremath{\epsilon} ) exists but that at order \ensuremath{\epsilon} the usual ${\mathit{w}}^{\mathrm{*}}$=0 fixed point is stable in the w direction. The associated correction-to-scaling exponent is ${\mathrm{\ensuremath{\omega}}}_{\mathit{w}}$=\ensuremath{\epsilon}+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$). It is shown that the ${\mathrm{\ensuremath{\varphi}}}_{\mathit{s}}^{3}$ term leads to a mixing of surface enhancement and surface field in the leading odd relevant surface scaling field. Furthermore, it is proven beyond perturbation theory that a (redundant) surface operator exists, that corresponds to a similar mixing in the leading even surface scaling field.The implications of this mixing for the critical behavior near the special surface transition are discussed both for the case of Ising ferromagnets as well as that of fluids. In the case of fluids, the mixing entails that quenched surface disorder of the random bond type is sufficient to prevent the occurrence of a (multicritical) special transition for bulk dimensions d\ensuremath{\le}3 A number of other redundant surface operators are identified, and a general redundancy criterion for surface operators (analogous to Wegner's for bulk operators) is given. Finally, a fixed point with ${\mathit{w}}^{\mathrm{*}}$=iO(\ensuremath{\surd}\ensuremath{\epsilon} ) is obtained, which describes the Yang-Lee edge singularities of an equivalent (d-1)-dimensional bulk system with long-range interactions. The associated critical exponents can be determined to all orders in \ensuremath{\epsilon}.

Physical review. B, Condensed matter•Oct 1, 1991•30 citations

Critical phenomena near free surfaces and defect planes

Theodore W. Burkhardt, Erich Eisenriegler

We discuss the universality classes of the local critical behavior of a bulk system near a free surface and near an internal plane of defects. A phase diagram and plausible renormalization-group flow properties for a general model which encompasses both cases are described qualitatively. Several of the crossover exponents are related to bulk critical exponents or semi-infinite surface critical exponents. The picture which is proposed accomodates systems with nonuniversal local critical exponents near a plane of defects, such as the two-dimensional Ising model near a defect line, in a natural way. Necessary conditions for nonuniversal local critical exponents and for logarithmic corrections are derived. The singularities of the energy densities near a free surface and near an internal defect plane on approaching the bulk critical temperature are analyzed.

Physical review. B, Condensed matter•Aug 1, 1981•59 citations

Dynamic surface critical behavior of systems with conserved bulk order parameter: Detailed RG analysis of the semi-infinite extensions of modelB with and without nonconservative surface terms

F. Wichmann, Hans Werner Diehl

arXiv:cond-mat/9410036•The European Physical Journal B•Jun 1, 1995•13 citations

Critical dynamics: a field-theoretical approach

R. Folk, G. Moser

We review the progress made in dynamic bulk critical behaviour in equilibrium in the last 25 years since the review of Halperin and Hohenberg. We unify the presentation of the theoretical background by restricting ourselves to the field-theoretic renormalization group method. The main results obtained in the different universality classes are presented. This contains the critical dynamics near the gas–liquid transition in pure fluids (model H), the plait point and consolute point in mixtures (model H � ), the superfluid transition in 4 He (model F) and 4 He– 3 He mixtures (model F � ), the Curie point (model J) and Neel point (model G) in Heisenberg magnets and the superconducting transition. In comparison with experimental results, it became clear that in most cases one has to consider apart from the universal asymptotic critical behaviour also the non-universal effective behaviour. Either because it turned out to be inevitable due to a small dynamical transient exponent inhibiting the system to reach the asymptotics (e.g., at the superfluid transition) or because one is interested in the region further away from the phase transition like in pure fluids and mixtures at their gas–liquid or demixing transition. The calculation of the critical dynamics is adequate in most cases only in two-loop order. We review these results and present the solution to unreasonable features found for some models. Thus, we consider model C where relaxational and diffusive dynamics are coupled and the scaling properties and the limit to a purely relaxational model (model A) have not been understood. In general for models where the order parameter couples to other conserved densities time scale ratios between the kinetic coefficients of the order parameter and the conserved densities play an important role. Their fixed-point values and the approach to the fixed point are changed considerably in two-loop order compared to their values in one-loop order. These considerations are relevant for the explanation of the dynamical critical shape functions of systems such as superfluid helium (model F) and the isotropic antiferromagnet (model G). As far as possible, the comparison of results obtained by the renormalization group theory with numerical simulations has been made.

Journal of Physics A Mathematical and General•May 31, 2006•150 citations
We investigate the problem of which universality classes of dynamic surface critical behavior exist for semi-infinite systems whose dynamic bulk critical behavior and static surface critical behavior are representative of a given dynamic bulk universality class and a given static surface universality class. To this end systems whose bulk dynamics are described by either model B or model A of Halperin, Hohenberg, and Ma are considered, where the dynamics are allowed to be modified near the surface as follows: In the case of model B, surface terms that locally break the conservation law for the order-parameter density \ensuremath{\varphi} are allowed; in the case of model A, \ensuremath{\varphi} is assumed to be locally conserved near the surface. Semi-infinite field-theory models are constructed that are (i) compatible with the requested bulk dynamics and the requirements of causality, detailed balance, and relaxation to thermal equilibrium and (ii) minimal in the sense that no redundant or irrelevant surface terms are included. The boundary conditions to which the surface terms of the action correspond are worked out. For model B it is shown that nonconservative surface terms are relevant. Their strength can be parametrized by a coupling constant c\ifmmode~\else˜\fi0{\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}\ifmmode~\else˜\fi0​\ensuremath{\ge}0 whose inverse plays the role of an extrapolation length for the auxiliary (Martin-Siggia-Rose) field \ensuremath{\varphi}\ifmmode \tilde{}\else ~{}\fi{}. Thus two distinct semi-infinite extensions of model B exist---one with c\ifmmode~\else˜\fi0{\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}\ifmmode~\else˜\fi0​>0 called model BA{\mathit{B}}_{\mathit{A}}BA​, and one with c\ifmmode~\else˜\fi0{\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{0}\ifmmode~\else˜\fi0​=0 called model BB{\mathit{B}}_{\mathit{B}}BB​---and each static surface universality class splits up into two dynamic surface universality classes.The static and dynamic surface critical exponents of these dynamic surface universality classes as well as the crossover exponent associated with the nonconservative surface term are shown to be expressible in terms of static bulk and surface exponents. Model BA{\mathit{B}}_{\mathit{A}}BA​ and its results should apply, e.g., to uniaxial ferromagnets whose rotational spin symmetry is broken in the vicinity of a surface plane. For model A the considered local modification of the dynamics near the surface is irrelevant in the renormalization-group sense and hence does not lead to new dynamic surface universality classes.
We study the surface critical behavior of semi-infinite systems belonging to the bulk universality class of the Ising model. Special attention is paid to the local behavior of experimentally relevant quantities such as the order parameter and the correlation function in the crossover regimes between different surface universality classes, where the surface field h1 and the temperature enhancement c can induce additional macroscopic length scales. Starting from the field-theoretical φ4 model and employing renormalization-group improved perturbation theory (ϵ-expansion), explicit results for the local behavior of the correlation and structure function (zero-loop) and the order parameter (one-loop) are derived. Supplementing earlier studies that focused on the special transition, we here pay particular attention to the situation where a large c suppresses the tendency to order in the surface (ordinary transition) but a surface field h1 generates a small surface magnetization m1. Our results are in good agreement with recent phenomenological considerations and Monte Carlo studies devoted to similar questions and, combined with the latter, provide a much more detailed understanding of the local properties of systems with weak surface fields.
We study the time evolution of classical spin systems with purely relaxational dynamics, quenched from $T\ensuremath{\gg}{T}_{c}$ to the critical point, in the semi-infinite geometry. Shortly after the quench, as in the bulk, a nonequilibrium regime governed by universal power laws is also found near the surface. We show for "ordinary" and "special" transitions that the corresponding critical exponents differ from their bulk values, but can be expressed via scaling relations in terms of known bulk and surface exponents. To corroborate our scaling analysis, we present perturbative ($\ensuremath{\epsilon}$-expansion) and Monte Carlo results.
The effects of weak symmetry-breaking surface fields on the surface critical behavior of semi-infinite systems with a scalar order parameter are investigated near a bulk critical point. Previous field-theoretic analyses in 4-\ensuremath{\epsilon} dimensions are generalized by incorporating an additional surface term w0{\mathit{w}}_{0}w0​${\mathrm{\ensuremath{\varphi}}}_{\mathit{s}}^{3}$ in the Hamiltonian. The mapping of microscopic models of the lattice-gas type on a continuum field theory is reconsidered, and we conclude that the cubic surface field w0{\mathit{w}}_{0}w0​ generically does not vanish both in the case of an Ising ferromagnet in an applied magnetic field H (where w0{\mathit{w}}_{0}w0​\ensuremath{\sim}H as H\ensuremath{\rightarrow}0) as well as in the case of critical adsorption of fluids onto a wall or an interface. Explicit results of a perturbative renormalization-group analysis are given, which show that a fixed point with nonvanishing cubic surface field w∗{\mathit{w}}^{\mathrm{*}}w∗=O(\ensuremath{\surd}\ensuremath{\epsilon} ) exists but that at order \ensuremath{\epsilon} the usual w∗{\mathit{w}}^{\mathrm{*}}w∗=0 fixed point is stable in the w direction. The associated correction-to-scaling exponent is ${\mathrm{\ensuremath{\omega}}}_{\mathit{w}}$=\ensuremath{\epsilon}+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$). It is shown that the ${\mathrm{\ensuremath{\varphi}}}_{\mathit{s}}^{3}$ term leads to a mixing of surface enhancement and surface field in the leading odd relevant surface scaling field. Furthermore, it is proven beyond perturbation theory that a (redundant) surface operator exists, that corresponds to a similar mixing in the leading even surface scaling field.The implications of this mixing for the critical behavior near the special surface transition are discussed both for the case of Ising ferromagnets as well as that of fluids. In the case of fluids, the mixing entails that quenched surface disorder of the random bond type is sufficient to prevent the occurrence of a (multicritical) special transition for bulk dimensions d\ensuremath{\le}3 A number of other redundant surface operators are identified, and a general redundancy criterion for surface operators (analogous to Wegner's for bulk operators) is given. Finally, a fixed point with w∗{\mathit{w}}^{\mathrm{*}}w∗=iO(\ensuremath{\surd}\ensuremath{\epsilon} ) is obtained, which describes the Yang-Lee edge singularities of an equivalent (d-1)-dimensional bulk system with long-range interactions. The associated critical exponents can be determined to all orders in \ensuremath{\epsilon}.
We discuss the universality classes of the local critical behavior of a bulk system near a free surface and near an internal plane of defects. A phase diagram and plausible renormalization-group flow properties for a general model which encompasses both cases are described qualitatively. Several of the crossover exponents are related to bulk critical exponents or semi-infinite surface critical exponents. The picture which is proposed accomodates systems with nonuniversal local critical exponents near a plane of defects, such as the two-dimensional Ising model near a defect line, in a natural way. Necessary conditions for nonuniversal local critical exponents and for logarithmic corrections are derived. The singularities of the energy densities near a free surface and near an internal defect plane on approaching the bulk critical temperature are analyzed.
We review the progress made in dynamic bulk critical behaviour in equilibrium in the last 25 years since the review of Halperin and Hohenberg. We unify the presentation of the theoretical background by restricting ourselves to the field-theoretic renormalization group method. The main results obtained in the different universality classes are presented. This contains the critical dynamics near the gas–liquid transition in pure fluids (model H), the plait point and consolute point in mixtures (model H � ), the superfluid transition in 4 He (model F) and 4 He– 3 He mixtures (model F � ), the Curie point (model J) and Neel point (model G) in Heisenberg magnets and the superconducting transition. In comparison with experimental results, it became clear that in most cases one has to consider apart from the universal asymptotic critical behaviour also the non-universal effective behaviour. Either because it turned out to be inevitable due to a small dynamical transient exponent inhibiting the system to reach the asymptotics (e.g., at the superfluid transition) or because one is interested in the region further away from the phase transition like in pure fluids and mixtures at their gas–liquid or demixing transition. The calculation of the critical dynamics is adequate in most cases only in two-loop order. We review these results and present the solution to unreasonable features found for some models. Thus, we consider model C where relaxational and diffusive dynamics are coupled and the scaling properties and the limit to a purely relaxational model (model A) have not been understood. In general for models where the order parameter couples to other conserved densities time scale ratios between the kinetic coefficients of the order parameter and the conserved densities play an important role. Their fixed-point values and the approach to the fixed point are changed considerably in two-loop order compared to their values in one-loop order. These considerations are relevant for the explanation of the dynamical critical shape functions of systems such as superfluid helium (model F) and the isotropic antiferromagnet (model G). As far as possible, the comparison of results obtained by the renormalization group theory with numerical simulations has been made.